Integrand size = 20, antiderivative size = 195 \[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {x^4}{8 b \pi }+\frac {\cos \left (b^2 \pi x^2\right )}{b^5 \pi ^3}+\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {3 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^5 \pi ^2}+\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {3 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^2 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \] Output:
-1/8*x^4/b/Pi+cos(b^2*Pi*x^2)/b^5/Pi^3+3*x*cos(1/2*b^2*Pi*x^2)*FresnelS(b* x)/b^4/Pi^2-3/2*FresnelC(b*x)*FresnelS(b*x)/b^5/Pi^2+3/8*I*x^2*hypergeom([ 1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)/b^3/Pi^2-3/8*I*x^2*hypergeom([1, 1],[3/2 , 2],1/2*I*b^2*Pi*x^2)/b^3/Pi^2+x^3*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^2/ Pi+1/4*x^2*sin(b^2*Pi*x^2)/b^3/Pi^2
\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx \] Input:
Integrate[x^4*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]
Output:
Integrate[x^4*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x], x]
Time = 1.07 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {7016, 3860, 3042, 3790, 15, 3042, 3777, 25, 3042, 3118, 7008, 3860, 3042, 3118, 7000}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 7016 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^3 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )^2dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3790 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int x^2dx^2}{2}-\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^4}{4}-\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^4}{4}-\frac {1}{2} \int x^2 \sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )dx^2}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {\int -\sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^4}{4}}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^4}{4}}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{\pi b^2}-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}\right )+\frac {x^4}{4}}{2 \pi b}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -\frac {3 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )+\frac {x^4}{4}}{2 \pi b}\) |
\(\Big \downarrow \) 7008 |
\(\displaystyle -\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )+\frac {x^4}{4}}{2 \pi b}\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle -\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )+\frac {x^4}{4}}{2 \pi b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )+\frac {x^4}{4}}{2 \pi b}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )+\frac {x^4}{4}}{2 \pi b}\) |
\(\Big \downarrow \) 7000 |
\(\displaystyle -\frac {3 \left (\frac {-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}}{\pi b^2}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}\right )}{\pi b^2}+\frac {x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (-\frac {x^2 \sin \left (\pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{\pi ^2 b^4}\right )+\frac {x^4}{4}}{2 \pi b}\) |
Input:
Int[x^4*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]
Output:
(-3*(-1/4*Cos[b^2*Pi*x^2]/(b^3*Pi^2) - (x*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x ])/(b^2*Pi) + ((FresnelC[b*x]*FresnelS[b*x])/(2*b) - (I/8)*b*x^2*Hypergeom etricPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi*x^2] + (I/8)*b*x^2*Hypergeometr icPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/(b^2*Pi)))/(b^2*Pi) + (x^3*Fres nelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (x^4/4 + (-(Cos[b^2*Pi*x^2]/(b^4 *Pi^2)) - (x^2*Sin[b^2*Pi*x^2])/(b^2*Pi))/2)/(2*b*Pi)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + ((f_.)*(x_))/2]^2, x_Symbol] :> Simp[1/2 Int[(c + d*x)^m, x], x] - Simp[1/2 Int[(c + d*x)^m*Cos[2*e + f *x], x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)], x_Symbol] :> Simp[FresnelC[b*x] *(FresnelS[b*x]/(2*b)), x] + (-Simp[(1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-2^(-1))*I*b^2*Pi*x^2], x] + Simp[(1/8)*I*b*x^2*HypergeometricP FQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2], x]) /; FreeQ[{b, d}, x] && EqQ[d^ 2, (Pi^2/4)*b^4]
Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-x ^(m - 1))*Cos[d*x^2]*(FresnelS[b*x]/(2*d)), x] + (Simp[(m - 1)/(2*d) Int[ x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Simp[1/(2*b*Pi) Int[x^(m - 1 )*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IG tQ[m, 1]
Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( m - 1)*Sin[d*x^2]*(FresnelS[b*x]/(2*d)), x] + (-Simp[1/(Pi*b) Int[x^(m - 1)*Sin[d*x^2]^2, x], x] - Simp[(m - 1)/(2*d) Int[x^(m - 2)*Sin[d*x^2]*Fre snelS[b*x], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m , 1]
\[\int x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )d x\]
Input:
int(x^4*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)
Output:
int(x^4*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)
\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:
integrate(x^4*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="fricas")
Output:
integral(x^4*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{4} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \] Input:
integrate(x**4*cos(1/2*b**2*pi*x**2)*fresnels(b*x),x)
Output:
Integral(x**4*cos(pi*b**2*x**2/2)*fresnels(b*x), x)
\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:
integrate(x^4*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="maxima")
Output:
integrate(x^4*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:
integrate(x^4*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="giac")
Output:
integrate(x^4*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
Timed out. \[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^4\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \] Input:
int(x^4*FresnelS(b*x)*cos((Pi*b^2*x^2)/2),x)
Output:
int(x^4*FresnelS(b*x)*cos((Pi*b^2*x^2)/2), x)
\[ \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelS}\left (b x \right )d x \] Input:
int(x^4*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)
Output:
int(x^4*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)